To determine the range of the relation given in the table, we need to identify the unique [tex]\( y \)[/tex]-values that correspond to any [tex]\( x \)[/tex]-values. Let’s go through the steps:
1. Extract the [tex]\( y \)[/tex]-values from the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 0 \\
\hline
-1 & 2 \\
\hline
0 & 4 \\
\hline
1 & 2 \\
\hline
2 & 0 \\
\hline
\end{array}
\][/tex]
From the table, the [tex]\( y \)[/tex]-values are: [tex]\( 0, 2, 4, 2, 0 \)[/tex].
2. Identify all unique [tex]\( y \)[/tex]-values:
Since we are looking for the range, we need to list all distinct [tex]\( y \)[/tex]-values. The unique values among [tex]\( 0, 2, 4, 2, 0 \)[/tex] are [tex]\( 0, 2, \)[/tex] and [tex]\( 4 \)[/tex].
3. Sort the unique [tex]\( y \)[/tex]-values (if needed):
In this context, it is often customary to present the range values in ascending order for clarity: [tex]\( 0, 2, 4 \)[/tex].
4. Write out the range:
Therefore, the range of the relation is:
[tex]\[
\{0, 2, 4\}
\][/tex]
Based on our analysis, the correct range of the relation is [tex]\(\{0, 2, 4\}\)[/tex]. This aligns with the first option provided.
